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Exam 3 \hspace{40mm}Name: \underline{\hspace{30mm}}

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Directions:  Answer each question to the best of your ability.  The point value of each question follows the question.  Calculators are allowed, but you must show appropriate work to receive full credit.  

Choose the best response for the following 10 questions. (4 pts each)

\begin{enumerate}
\item When taking a slice with the width $dx$, which of the following statements are true?
\begin{enumerate}
\item The slice is parallel to the $x$-axis
\item The slice is perpendicular to $x$-axis
\item The slice is perpendicular to the $y$-axis
\item The orientation of the slice cannot be determined
\end{enumerate}

\item Which type of slice will you use to find the volume of revolution of $f(x)=x^2-x+1$ on [0,4] around the $x$-axis?
\begin{enumerate}
\item Circle or Disk
\item Washer
\item Cylindrical Shell
\item None of the above
\end{enumerate}

\item Which is the correct integral to find the volume of revolution of the area between $f(x)=x^2$ and $g(x)=x$ on [0,1] revolved around the $x$-axis?
\begin{enumerate}
\item $\int_0^1\,\pi(x^2)^2\,dx$
\item $\int_0^1\,\pi (x^4-x^2)\,dx$
\item $\int_0^1\,2\pi x(x-x^2)\,dx$
\item $\int_0^1\,\pi (x^2-x^4)\,dx$
\end{enumerate}

\item Which of the following integrals will \underline{NOT} find the volume of revolution of the area between $f(x)=x^2$ and $g(x)=x$ on [0,1] revolved around the $y$-axis?
\begin{enumerate}
\item $\int_0^1\,2\pi x(x-x^2)\,dx$
\item $\int_0^1\,\pi(y-y^2)\,dy$
\item $\int_0^1\,\pi (x^2-x^4)\,dx$
\item All three integrals will find the correct area
\end{enumerate}

\item To find $\int \,x\,e^x$, which of the following methods will produce the correct answer
\begin{enumerate}
\item Substitution where O.F. = $e^u$ and I.F. = $u=x$
\item Integration by parts where $u=e^x$ and $v'=x$
\item Integration by parts where $u=x$ and $v'=e^x$
\item None of these previous methods will produce the correct answer
\end{enumerate}
\newpage
\item A square pyramid has a base area of 144 $m^2$ and a height of 24 $m$.  If you slice the pyramid parallel to the base, which is the correct formula for the volume of a slice in terms of the height, $h$, from the apex of the pyramid?
\begin{enumerate}
\item $V=\frac{1}{4}h^2\,d h$
\item $V=\frac{1}{2}h^2\,d h$
\item $V=2h^2\,dh$
\item $V=4h^2\,d h$
\end{enumerate}
\item We have learned two formulas to find the work done on an object, one using vectors and calculating the dot product and the other finding the definite integral of the force function with respect to position.  What condition would compel us to use the second formula?
\begin{enumerate}
\item The displacement vector has a negative magnitude
\item The force is not constant
\item The force acts perpendicular to the displacement
\item The force acts parallel to the displacement
\end{enumerate}
\item To find $\int\,\sin x \sqrt{\cos x}$, which of the following methods will produce the correct answer?
\begin{enumerate}
\item Integration by Parts where $u=\sin x$ and $v'=\sqrt{cos x}$
\item Substitution where O.F.= $\sqrt{u}$ and I.F.= $ u=\cos x$
\item Integration by Parts where $v'=\sin x$ and $u=\sqrt{cos x}$
\item None of the previous three methods will provide the correct answer
\end{enumerate}
\item Given the population density in terms of distance from a city's downtown is $\delta (r)$, which is the correct integral to find the total population living with 15 $mi$ in all directions from downtown?
\begin{enumerate}
\item $\int_0^{15}\, 2\pi \delta (r)\, dx$
\item $\int_0^{15}\, \delta (r)\, dr$
\item $\int_0^{15}\, \pi (\delta (r))^2\, dx$
\item None of the previous three integrals will find the correct total population
\end{enumerate}
\item A 100$lb$ weight is suspended from the roof of a building by a 50$ft$ chain which weighs 2$lb/ft$.  Which integral will find the total work done to lift the weight and chain onto the roof?
\begin{enumerate}
\item $\int_0^{50}\, 100\, dx$
\item $\int_0^{50}\, 100+2x\, dx$
\item $\int_0^{50}\, 100+2(50-x)\, dx$
\item The work cannot be determined as we need to know the height of the building first.
\end{enumerate}
\newpage



\item Find the area between $f(x)=x^2$ and $g(x)=4x$ (Hint: find where the two functions intersect) (6 pts)
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\item Find the volume of revolution of the area between $f(x)=x^2$ and $g(x)=4x$ revolved around the $x$-axis (6 pts)
\vfill
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\item Find the volume of revolution of the area between $f(x)=x^2$ and $g(x)=4x$ revolved around the $y$-axis (6 pts)
\vfill
\vfill
\newpage


\item Find the volume of a square pyramid which has a base area of 144 $m^2$ and a height of 24 $m$ using a Riemann sum converted to a definite integral. (6 pts)
\vfill
\item Given the base of the pyramid is parallel to the $yz$-plane, and the apex of the pyramid is at the origin, find the coordinates of the center of mass of the pyramid. (6 pts)
\vfill
\newpage
\item A metal object fits the shape of the function $y=x^2$ revolved around the $x$-axis bounded by the interval [0,10].  If the function for the density of the object given the distance from the narrow end is $\delta (x) = \frac{100}{x^2}$ $kg/m^3$, find:
\begin{enumerate}
\item The total mass of the object (6 pts)
\vfill
\item The center of mass of the object (6 pts)
\vfill
\end{enumerate}
\newpage

\item A 1000 $lb$ block of ice is lifted 75 $ft$ into the air.  If the outside temperature causes the block to melt causing a loss of 3 $lb$ for every foot lifted, what is the total work done to lift the block of ice? (6 pts)
\vfill
\item A inverted cone-shaped tank is filled with water.  The height of the cone is 20 feet and the radius at the top is 5 feet.  How much work is necessary to pump all the water up out of the tank? (6 pts)
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\item A dam has a trapezoidal shape with a top width of 500 feet, bottom width of 300 feet, and height of 100 feet.  Assuming the dam is at capacity, what is the force of the water pressing against the dam? (6 pts)
\vfill
\end{enumerate}
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